m-injective modules and prime m-ideals - Department of ...

(2) The subcategory σ[M]σ[M] is the full subcategory of R–Mod that contains allRX isomorphic to a submodule of an M-generated module.σ[M] = R–Mod iff R belongs to σ[M] iff M is faithful andfinitely annihilated.σ[M] is closed under homomorphic images, submodules,and direct sums. It has pullbacks, pushouts, a generator,and products. Injective modules and injective envelopesexist.Definition 1.2. The submodule N ⊆ M is called an M-ideal if there is a class C in σ[M] with N = Ann M (C).Proposition [Bican, Jambor, Kepka, Němec: 1977] Thefollowing are equivalent for a submodule N ⊆ M.(1) N is an M-ideal;(2) there is a radical ρ of R–Mod with N = ρ(M);(3) g(N) = (0) for all g ∈ Hom R (M, (M/N));(4) N = Ann M (M/N).